Fractals and Disordered Systems

Transcription

1 Fractals and Disordered Systems Springer Berlin Heidelberg New York Barcelona Budapest Hong Kong London Milan Paris Santa Clara Singapore Tokyo

2 Armin Bunde Shlomo Havlin (Eds.) Fractals and Disordered Systems Second Revised and Enlarged Edition With 165 Figures and 10 Color Plates Springer

3 Professor Dr. Armin Bunde Institut fur Theoretische Physik Universitat Giessen Heinrich-Buff-Ring 16 D Giessen Germany Professor Dr. Shlomo Havlin Department of Physics Bar-Han University RamatGan Israel ISBN-13: e-isbn-13: DOl: / Library of Congress Cataloging-in-Publication Data. Fractals and disordered systems 1 Armin Bunde, Shlomo Havlin (eds.). - 2nd ed. p. cm. Includes bibliographical references and index. (alk. paper). 1. Fractals. 2. Chaotic behavior in systems. 3. Order-disorder models. I. Bunde, Armin, II. Havlin, Shlomo. QA F '13-dc This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse ofillustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereofis permitted only under the provisions ofthe German Copyright Law of September 9,1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. Springer-Verlag Berlin Heidelberg 1991, 1996 Softcover reprint of the hardcover 2nd edition 1996 The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Camera-ready copy from the authorsleditors using a Springer TEX macro package SPIN: / Printed on acid-free paper

4 Preface to the Second Edition The growing interest in fractals and disordered systems made a second edition of this book necessary. We took advantage of this situation to make, together with the other authors, major revisions in order to include new trends and achievements in this fast-developing field. Many sections were revised completely and several sections and subsections were added. For example, H. Eugene Stanley added, in Chap. 1, new sections on DLA (glove and skeleton), multiscaling and self-affine surfaces. In Chaps. 2 and 3, we revised completely the sections dealing with the important chemical distance concept and added sections on branched polymers and elasticity. In Chap. 5, Hans Herrmann added a section on hydraulic fracture, which is of high technological importance. Bernhard Sapoval revised Chap. 6 significantly by considering transfer across irregular surfaces in a more general approach. In Chap. 9, Dietrich Stauffer added a section on recent biologically motivated developments of cellular automata. Benoit Mandelbrot added ten pages to Chap. 10, where he presented new ideas and results on multifractal and Minkowski measures. Also, all the tables and references have been updated. We wish to thank Dr. Hans K6lsch and Petra Treiber from Springer-Verlag for their very helpful and constructive cooperation. Giessen Ramat-Gan September 1995 Armin Bunde Shlomo Havlin

5 Preface to the First Edition Disordered structures and random processes that are self-similar on certain length and time scales are very common in nature. They can be found on the largest and the smallest scales: in galaxies and landscapes, in earthquakes and fractures, in aggregates and colloids, in rough surfaces and interfaces, in glasses and polymers, in proteins and other large molecules. Owing to the wide occurrence of self-similarity in nature, the scientific community interested in this phenomenon is very broad, ranging from astronomers and geoscientists to material scientists and life scientists. Among the major achievements in recent years that have strongly influenced our understanding of structural disorder and its formation by random processes are the fractal concepts pioneered by B. B. Mandelbrot. Fractal geometry is a mathematical language used to describe complex shapes and is particularly suitable for computers because of its iterative nature. The field of fractals and disordered systems is developing very rapidly, and there exists a large gap between the knowledge presented in advanced textbooks and the research front. The aim of this book is to fill this gap by introducing the reader to the basic concepts and modern techniques in disordered systems, and to lead him to the forefront of current research. The book consists of ten chapters, written with a uniform notation, with cross references in each chapter to related subjects in other chapters. In each chapter emphasis has been placed on connections between theory and experiment. A special chapter (Chap. 8) discusses experimental studies of fractal systems. In the first chapter, H. E. Stanley introduces relevant fractal concepts, from self-similarity to multifractality. Several fractal prototypes such as random walks and diffusion limited aggregation (DLA) are discussed in detail in order to demonstrate the various concepts and their utility. The possibility that DLA is a prototype fractal describing many different phenomena in biology, chemistry and physics, is stressed.

6 VIn Preface to the First Edition The next two chapters, written by A. Bunde and S. Havlin, present a detailed description of the static (Chap. 2) and dynamical (Chap. 3) properties of percolation systems. Percolation is a common model for disordered systems, and its relation to the fractal concept is emphasized. Many theoretical and numerical approaches such as scaling theories, series expansions, renormalization, and exact enumeration of random walks are discussed. In Chap. 4, A. Aharony applies the ideas of fractals and multi fractals to several models of random growth processes; examples are the Eden model, invasion percolation and general aggregation processes. The relation between DLA and electrodeposition processes, viscous fingering, dielectric breakdown, and the Laplacian growth processes is discussed. Chapter 5 by H. J. Herrmann discusses microscopic models for the formation and the complex structure of fractures, which represent a subject of great technological importance. The microscopic modeling of fractures is quite a new topic. Previous work was mainly devoted to phenomenological properties of fractures. In Chap. 6, B. Sapoval studies the physical and chemical properties of irregular surfaces such as electrodes, membranes and catalysts, interest in which has been revived by the concept of fractal geometry. Many natural or industrial processes take place through surfaces or across the interfaces between two media. For example tree roots exchange water and inorganic salts with the earth through the surface of the roots; the larger the interface area, the more effective the process. Porous fractals are objects that have this property. This is followed in Chap. 7 by several models for rough surfaces and interfaces presented by J. F. Gouyet, M. Rosso and B. Sapoval. The authors begin with a general characterization of rough surfaces, followed by deposition models and diffusion fronts and ending with fluid-fluid interfaces, membranes, and tethered surfaces. In Chap. 8, J. K. Kjems presents a broad review on the experimental techniques used to study fractal features in disordered systems. The chapter deals with the questions of how fractal realizations can be made, how to measure the fractal dimension, and how to measure physical properties of fractals. In Chap. 9, D. Stauffer provides systematic approaches as well as computational techniques for the study of cellular automata where the fractal concept is relevant, including the Wolfram characterization and, among others, the Kaufman model for genetics. In the last chapter, B. B. Mandelbrot and C. J. G. Evertsz focus on exactly self-similar left-sided multifractals, which seem to be relevant to fully developed turbulence and DLA. The authors present explicit examples of multiplicative multifractals having the property that the multifractal scaling relation fails to hold, either for small enough negative moments, or for high enough positive moments. We wish to thank the authors for their cooperation and many of our colleagues, in particular M. A. Denecke, R. Nossal, H. E. Roman, and H. Taitel-

7 Preface to the First Edition IX baum for many useful discussions. We kindly acknowledge the help of J. D. Chen, F. Grey, A. Hansen, C. Kolb, R. A. Masland, P. Meakin, C. Roessler, S. Roux, S. Schwarzer, B. L. Thus, D. A. Weitz, and S. Wolfram, who produced the beautiful color figures appearing in the book. We hope that the book can be used as a first textbook for graduate students, for teachers at universities for preparing regular courses or seminars, and for researchers who encounter fractals and disordered systems. Hamburg Washington June 1991 Armin Bunde Shlomo Havlin

8 Contents 1 Fractals and Multifractals: The Interplay of Physics and Geometry By H. Eugene Stanley (With 30 Figures) Introduction.... Nonrandom Fractals... : Random Fractals: The Unbiased Random Walk.... The Concept of a Characteristic Length.... Functional Equations and Fractal Dimension.... An Archetype: Diffusion Limited Aggregation.... DLA: Fractal Properties.... DLA: Multifractal Properties General Considerations "Phase Transition" in 2d DLA The Void-Channel Model of 2d DLA Growth Multifractal Scaling of 3d DLA.... Scaling Properties of the Perimeter of 2d DLA: The "Glove" Algorithm Determination of the C Perimeter The C Gloves Necks and Lagoons.... Multiscaling.... The D LA Skeleton.... Applications of DLA to Fluid Mechanics Archetype 1: The Ising Model and Its Variants Archetype 2: Random Percolation and Its Variants Archetype 3: The Laplace Equation and Its Variants

9 XII Contents Applications of DLA to Dendritic Growth Fluid Models of Dendritic Growth Noise Reduction Dendritic Solid Patterns: "Snow Crystals" Dendritic Solid Patterns: Growth of NH4Br.... Other Fractal Dimensions The Fractal Dimension dw of a Random Walk The Fractal Dimension dmin == IIi) of the Minimum Path Fractal Geometry of the Critical Path: "Volatile Fractals" Surfaces and Interfaces Self-Similar Structures Self-Affine Structures LA Appendix: Analogies Between Thermodynamics and Multifractal Scaling References Percolation I By Armin Bunde and Shlomo Havlin (With 24 Figures) 2.1 Introduction Percolation as a Critical Phenomenon Structural Properties Exact Results One-Dimensional Systems The Cayley Tree Scaling Theory Scaling in the Infinite Lattice Crossover Phenomena Finite-Size Effects Related Percolation Problems Epidemics and Forest Fires Kinetic Gelation Branched Polymers Invasion Percolation Directed Percolation Numerical Approaches Hoshen-Kopelman Method Leath Method Ziff Method... 98

10 Contents XIII 2.8 Theoretical Approaches Deterministic Fractal Models Series Expansion Small-Cell Renormalization Potts Model, Field Theory, and E Expansion A Appendix: The Generating Function Method References Percolation II By Shlomo Havlin and Armin Bunde (With 20 Figures) 3.1 Introduction Anomalous 'Ifansport in Fractals Normal 'Ifansport in Ordinary Lattices 'Ifansport in Fractal Substrates 'Ifansport in Percolation Clusters Diffusion in the Infinite Cluster Diffusion in the Percolation System.... Conductivity in the Percolation System Transport in Two-Component Systems.... Elasticity in Two-Component Systems Fractons Elasticity Vibrations of the Infinite Cluster Vibrations in the Percolation System Quantum Percolation ac Transport Lattice-Gas Model Equivalent Circuit Model Dynamical Exponents Rigorous Bounds Numerical Methods Series Expansion and Renormalization Methods Continuum Percolation Summary of 'Ifansport Exponents Multifractals Voltage Distribution Random Walks on Percolation 155

11 XIV Contents 3.8 Related 1fansport Problems Biased Diffusion Dynamic Percolation The Dynamic Structure Model of Ionic Glasses Trapping and Diffusion Controlled Reactions References Fractal Growth By Amnon Aharony (With 4 Figures) 4.1 Introduction Fractals and Multifractals Growth Models Eden Model Percolation Invasion Percolation Laplacian Growth Model Diffusion Limited Aggregation Dielectric Breakdown Model Viscous Fingering Biological Growth Phenomena Aggregation in Percolating Systems Computer Simulations Viscous Fingers Experiments Exact Results on Model Fractals Crossover to Homogeneous Behavior Crossover in Dielectric Breakdown with Cutoffs Is Growth Multifractal? Conclusion.... References Fractures By Hans J. Herrmann (With 18 Figures) 5.1 Introduction Some Basic Notions of Elasticity and Fracture Phenomenological Description Elastic Equations of Motion

12 Contents XV 5.3 Fracture as a Growth Model Formulation as a Moving Boundary Condition Problem Linear Stability Analysis Modelisation of Fracture on a Lattice Lattice Models Equations and Their Boundary Conditions Connectivity The Breaking Rule The Breaking of a Bond Summary Deterministic Growth of a Fractal Crack Scaling Laws of the Fracture of Heterogeneous Media Hydraulic Fracture Conclusion References Transport Across Irregular Interfaces: Fractal Electrodes, Membranes and Catalysts By Bernard Sapoval (With 8 Figures) 6.1 Introduction The Electrode Problem and the Constant Phase Angle Conjecture The Diffusion Impedance and the Measurement of the Minkowski-Bouligand Exterior Dimension The Generalized Modified Sierpinski Electrode A General Formulation of Laplacian Transfer Across Irregular Surfaces Electrodes, Roots, Lungs, Fractal Catalysts Summary References Fractal Surfaces and Interfaces By Jean-Frangois Gouyet,Michel Rosso and Bernard Sapoval (With 27 Figures) 7.1 Introduction Rough Surfaces of Solids

13 XVI Contents Self-Affine Description of Rough Surfaces Growing Rough Surfaces: The Dynamic Scaling Hypothesis Deposition and Deposition Models Fractures Diffusion Fronts: Natural Fractal Interfaces in Solids Diffusion Fronts of Noninteracting Particles Diffusion Fronts in d = Diffusion Fronts of Interacting Particles Fluctuations in Diffusion Fronts Fractal Fluid-Fluid Interfaces Viscous Fingering Multiphase Flow in Porous Media Membranes and Tethered Surfaces Conclusions References Fractals and Experiments By Jorgen K. Kjems (With 18 Figures) 8.1 Introduction Growth Experiments: How to Make a Fractal The Generic DLA Model Dielectric Breakdown Electrodeposition Viscous Fingering Invasion Percolation Colloidal Aggregation Structure Experiments: How to Determine the Fractal Dimension Image Analysis Scattering Experiments Sacttering Formalism Physical Properties Mechanical Properties Thermal Properties Outlook References

14 Contents XVII 9 Cellular Automata By Dietrich Stauffer (With 6 Figures) 9.1 Introduction A Simple Example The Kauffman Model Classification of Cellular Automata Recent Biologically Motivated Developments A Appendix A.1 Q2R Approximation for Ising Models A.2 Immunologically Motivated Cellular Automata A.3 Hydrodynamic Cellular Automata References Exactly Self-similar Left-sided Multifractals By Benoit B. Mandelbrot and Carl J.G. Evertsz with new Appendices Band C by Rudolf H. Riedi and Benoit B. Mandelbrot (With 10 Figures) 10.1 Introduction Two Distinct Meanings of Multifractality "Anomalies" Nonrandom Multifractals with an Infinite Base Left-sided Multifractality with Exponential Decay of Smallest Probability A Gradual Crossover from Restricted to Left-sided Multifractals Pre-asymptotics Sampling of Multiplicatively Generated Measures by a Random Walk An "Effective" j(o'.) Miscellaneous Remarks Summary A Details of Calculations and Further Discussions A.1 Solution of (10.2) A.2 The Case O'.min = B Multifractal Formalism for Infinite Multinomial Measures, by R.H. Riedi and B.B. Mandelbrot C The Minkowski Measure and Its Left-sided f(o'.), by B.B. Mandelbrot

15 XVIII Contents 10.C.1 The Minkowski Measure on the Interval [0,1] C.2 The Functions f(a) and fe(a) of the Minkowski Measure C.3 Remark: On Continuous Models as Approximations, and on "Thermodynamics" C.4 Remark on the Role of the Minkowski Measure in the Study of Dynamical Systems. Parabolic Versus Hyperbolic Systems C.5 In Lieu of Conclusion References Subject Index 401

16 List of Contributors Amnon Aharony School of Physics and Astronomy, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv 69978, Israel and Institute of Physics, University of Oslo, Norway Armin Bunde Institut fur Theoretische Physik, Justus-Liebig-Universitat, Heinrich-Buff-Ring 16, D Giessen, Germany Carl J. G. Evertsz Fachbereich Mathematik, UniversiUit Bremen, D Bremen, Germany Jean-Frangois Gouyet Laboratoire de Physique de la Matiere Condensee, Ecole Poly technique, F Palaiseau, France

17 XX List of Contributors Shlomo Havlin Department of Physics, Bar-Ilan University, Ramat Gan, Israel, and Center of Polymer Studies and Department of Physics, Boston University, Boston, MA 02215, USA Hans J. Herrmann L.H.M.P, E.S.P.C.I., F Paris, France and Institut fur Computeranwendungen I, Universitiit Stuttgart, D Stuttgart, Germany J0rgen K. Kjems Ris0 National Laboratory, DK-4000 Roskilde, Denmark Benoit B. Mandelbrot Physics Department, IBM T.J. Watson Research Center, Yorktown Heights, NY 10598, USA and Mathematics Department, Yale University, Box 2155 Yale Station, New Haven, CT 06520, USA Rudolf H. Riedi Mathematics Department, Yale University, Box 2155 Yale Station, New Haven, CT 06520, USA Michel Rosso Laboratoire de Physique de la Matiere Condensee, Ecole Poly technique, F Palaiseau, France

18 List of Contributors XXI Bernard Sapoval Laboratoire de Physique de la Matiere Condensee, Ecole Poly technique, F Palaiseau, France H. Eugene Stanley Center of Polymer Studies and Department of Physics, Boston University, Boston, MA 02215, USA Dietrich Stauffer Institut fur Theoretische Physik, Universitat Kaln, D Kaln, Germany

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